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Multiply the following using the vertical multiplication method:

[tex]\[
\begin{array}{l}
3x^2 - 5x + 1 \\
\times (x^2 + 2x + 4) \\
\hline
\end{array}
\][/tex]

A. [tex]\(3x^4 - x^3 + 3x^2 + 18x + 4\)[/tex]

B. [tex]\(3x^4 + x^3 + 10x^2 + x + 4\)[/tex]

C. [tex]\(3x^4 + x^3 + 3x^2 - 18x + 4\)[/tex]

D. [tex]\(3x^4 + x^3 + 10x^2 - 18x + 4\)[/tex]

Sagot :

GinnyAnswer
To multiply the given polynomials [tex]\( 3x^2 - 5x + 1 \)[/tex] and [tex]\( x^2 + 2x + 4 \)[/tex] using the vertical multiplication method, follow these steps:

### Step-by-Step Process

1. List the Polynomials:
- First Polynomial: [tex]\( 3x^2 - 5x + 1 \)[/tex]
- Second Polynomial: [tex]\( x^2 + 2x + 4 \)[/tex]

2. Multiply Each Term Independently:

- Multiply [tex]\( 3x^2 \)[/tex] by Each Term in [tex]\( x^2 + 2x + 4 \)[/tex]:
- [tex]\( 3x^2 \cdot x^2 = 3x^4 \)[/tex]
- [tex]\( 3x^2 \cdot 2x = 6x^3 \)[/tex]
- [tex]\( 3x^2 \cdot 4 = 12x^2 \)[/tex]

- Multiply [tex]\(-5x\)[/tex] by Each Term in [tex]\( x^2 + 2x + 4 \)[/tex]:
- [tex]\(-5x \cdot x^2 = -5x^3 \)[/tex]
- [tex]\(-5x \cdot 2x = -10x^2 \)[/tex]
- [tex]\(-5x \cdot 4 = -20x \)[/tex]

- Multiply [tex]\( 1 \)[/tex] by Each Term in [tex]\( x^2 + 2x + 4 \)[/tex]:
- [tex]\( 1 \cdot x^2 = x^2 \)[/tex]
- [tex]\( 1 \cdot 2x = 2x \)[/tex]
- [tex]\( 1 \cdot 4 = 4 \)[/tex]

3. Combine All the Results:
- [tex]\( 3x^4 \)[/tex]
- [tex]\( 6x^3 + (-5x^3) = x^3 \)[/tex]
- [tex]\( 12x^2 + (-10x^2) + x^2 = 3x^2 \)[/tex]
- [tex]\(-20x + 2x = -18x \)[/tex]
- [tex]\( 4 \)[/tex]

4. Sum the Combined Results:
- [tex]\( 3x^4 + x^3 + 3x^2 - 18x + 4 \)[/tex]

### Conclusion
Therefore, the final result of multiplying [tex]\( 3x^2 - 5x + 1 \)[/tex] by [tex]\( x^2 + 2x + 4 \)[/tex] using the vertical multiplication method is:

[tex]\[ 3x^4 + x^3 + 3x^2 - 18x + 4 \][/tex]

Thus, the correct answer is:

C. [tex]\( 3x^4 + x^3 + 3x^2 - 18x + 4 \)[/tex]
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